Stages and main tasks of the century-long control theory and system identification development. Part IV. Methods and problems of designing robust control systems

The article provides a review of the most important methods and problems in the design of robust discrete control systems. In this case, the main attention was focused on the problems of suppressing limited external disturbances, information about which is presented only in the form of a limitation on their maximum value. The use of invariant ellipsoids is considered as the first mathematical apparatus for describing the characteristics of the influence of external disturbances on the trajectory of motion of dynamic systems. Theorems on the representation of invariant ellipsoids in the form of linear matrix inequalities (LMI) are formulated, which are further used to synthesize discrete state controllers that suppress external disturbances. The solution to a more general problem of robust suppression of limited disturbances based on the use of LMI in the presence of system uncertainties in the parameters of the mathematical model of the control object is considered. The use of H∞-control theory is considered as a second mathematical apparatus for suppressing external l2-limited external disturbances. In this case, the optimality criterion consists in minimizing the maximum ratio of the l2-norm of the vector of output stabilized coordinates to the l2-norm of the vector of input disturbances. The problem is solved by reducing it to the problem of robust control of a discrete dynamic system in space H∞ based on the Two-Ricatti approach. The standard H∞-optimization problem is also considered.